A financial market with interacting investors: does an equilibrium exist?

Abstract

While trading on a financial market, the agents we consider take the performance of their peers into account. By maximizing individual utility subject to investment constraints, the agents may ruin each other even unintentionally so that no equilibrium can exist. However, when the agents are willing to waive little expected utility, an approximated equilibrium can be established. The study of the associated backward stochastic differential equation (BSDE) reveals the mathematical reason for the absence of an equilibrium. Presenting an illustrative counterexample, we explain why such multidimensional quadratic BSDEs may not have solutions despite bounded terminal conditions and in contrast to the one-dimensional case.

A stability approach for solving multidimensional quadratic BSDEs

“…Let us remark that, in addition to its intrinsic mathematical interest, this question is important due to many applications of such equations. We can mention for example following applications: nonzero-sum risk-sensitive stochastic differential games in [EKH03,HT16], financial market equilibrium problems for several interacting agents in [ET15,FDR11,Fre14,BLDR15], financial price-impact models in [KP16b,KP16a], principal agent contracting problems with competitive interacting agents in [EP16], stochastic equilibria problems in incomplete financial markets [KXŽ15,XŽ16] or existence of martingales on curved spaces with a prescribed terminal condition [Dar95]. Let us note that moving from the scalar framework to the multidimensional one is quite challenging since tools usually used when d = 1, like monotone convergence or Girsanov transform, can no longer be used when d > 1.…”

Multi-dimensional backward stochastic differential equations of diagonally quadratic generators

“…Frei and dos Reis [5] constructed a counterexample to show that multi-dimensional quadratic BSDE (1.1) might fail to have a global solution (Y, Z) on [0, T ] such that Y is essentially bounded, which illustrates the difficulty of the quadratic part contributing to the underlying scalar generator as an unbounded process-the exponential of whose time-integral is likely to have no finite expectation. Very recently, Cheridito and Nam [4] addressed a special system of quadratic BSDEs in the Markovian context, which arises from the equilibrium problem with interacting agents in a financial market (see Frei and dos Reis [5] for further descriptions). Neither global nor local (in time) positive results are found in the literature for solvability of multidimensional quadratic BSDEs.…”

Mean field and n‐agent games for optimal investment under relative performance criteria

“…and that (54) yields (56). 25 (ii) Suppose δ = 1. It is easily checked that the solution v of (55) is given by…”